Answer
Using Eq. (8) and Eq. (9) we obtain
$E = - \frac{{GMm}}{{2R}}$
Work Step by Step
The total energy of a planet of mass $m$ orbiting a sun is given by Eq. (8):
(8) ${\ \ \ \ }$ $E = \frac{1}{2}m{v^2} - \frac{{GMm}}{{||{\bf{r}}||}}$
Since the orbit is circular of radius $R$, Eq. (8) becomes
$E = \frac{1}{2}m{v^2} - \frac{{GMm}}{R}$
From Exercise 9, the velocity of the planet in a circular orbit of radius $R$ is $v = \sqrt {\frac{{GM}}{R}} $. Thus,
$E = \frac{1}{2}m\left( {\frac{{GM}}{R}} \right) - \frac{{GMm}}{R}$
$E = - \frac{{GMm}}{{2R}}$
